FIG. 17 shows a conventional voltage subtractor/adder circuit described in a publication (IEEE Journal of Solid-State Circuits, Vol. CAS-32, No. 11, pp. 1097-1104, November 1985). The circuit of FIG. 17 comprises two sets of MOS differential pairs. One of the MOS differential pairs comprises MOS transistors M1 and M2, and the other of the MOS differential pairs comprises MOS transistors M3 and M4. Each of the MOS differential pairs is driven by a tail current Iss.
In the voltage subtractor/adder circuit shown in FIG. 17, voltages V1 and V2 are applied to the gates of the transistors M1 and M4, respectively, of the two sets of MOS differential pairs. Both the transistors M2 and M3 are diode-coupled and are driven by a common constant current source (Iss).
Here, with respect to the two sets of MOS differential pairs, tail current values of respective MOS differential pairs and a current value of a constant current source which drives the diode-coupled transistors M2 and M3 are all the same. Therefore, the following formulas are obtained.ID1+ID2=Iss  (1)ID3+ID4=Iss  (2)ID2+ID3=Iss  (3)where, ID1, ID2, ID3 and ID4 designates drain currents of the transistors M1, M2, M3 and M4, respectively. Therefore, the following relations are also obtained.ID1=ID3  (4)ID2=ID4  (5)
That is, since the currents flowing through the transistors M1 and M2 are equal to the currents flowing through the transistors M3 and M4, respectively, the differential input voltages of the two sets of MOS differential pairs become equal to each other. Therefore, assuming that a common gate potential of the diode-coupled transistors M2 and M3 is V0, the following relation exists.V1−V0=V0−V2  (6)That is, the following formula is obtained.                               V          0                =                                            V              1                        +                          V              2                                2                                    (        7        )            From this formula, it can be seen that the circuit shown in FIG. 17 functions as a voltage adder circuit. In this case, the differential input voltage of each of the MOS differential pairs becomes as follows:                                           V            1                    -                      V            0                          =                                            V              0                        -                          V              2                                =                                                    V                1                            -                              V                2                                      2                                              (        8        )            
Next, drain currents ID1 and ID4 of the transistors M1 and M4, respectively, of the MOS differential pairs will be derived.
Neglecting the body effect and the channel length modulation, and assuming that the relationship between a drain current and a gate-source voltage of a MOS transistor operating in saturation region follows the square-law, the drain current of a MOS transistor can be represented as follows:ID=β(VGS−VTH)2 (VGS≧VTH)  (9a)ID=0 (VGS≦VTH)  (9b)Here, β=μ(COX/2)(W/L) is a transconductance parameter, μ is an effective mobility of carrier, COX is capacitance of a gate oxide film per unit area, W is a gate width, L is a gate length, and VTH is the threshold voltage of a MOS transistor.
Assuming that the MOS transistors are matched well, the drain currents of the transistors M1 and M4 become as follows:                     {                                                                                                                        I                      D4                                        =                                                                  1                        2                                            ⁢                                              {                                                                              I                            SS                                                    +                                                      β                            ⁢                                                                                          V                                i                                                            2                                                        ⁢                                                                                                                                                                2                                    ⁢                                                                          I                                      SS                                                                                                        β                                                                -                                                                                                      V                                    i                                    2                                                                    4                                                                                                                                                                    }                                            ⁢                                              (                                                                                                                                        V                              i                                                                                                            ≤                                                      2                            ⁢                                                                                                                            I                                  SS                                                                β                                                                                                                                    )                                            ⁢                                                                                           ⁢                                              (10a)                                                                              ⁢                                                                                                                                                                                   I                    D4                                    =                                                            1                      2                                        ⁢                                          I                      SS                                        ⁢                                          sgn                      ⁡                                              (                                                  V                          i                                                )                                                              ⁢                                          (                                                                                                                              V                            i                                                                                                    ≥                                                  2                          ⁢                                                                                                                    I                                SS                                                            β                                                                                                                          )                                        ⁢                                                                                   ⁢                                          (10b)                                                                                                    ⁢                                          ⁢                      {                                                                                                      I                      D1                                        =                                                                  1                        2                                            ⁢                                              {                                                                              I                            SS                                                    +                                                      β                            ⁢                                                                                          V                                i                                                            2                                                        ⁢                                                                                                                                                                2                                    ⁢                                                                          I                                      SS                                                                                                        β                                                                -                                                                                                      V                                    i                                    2                                                                    4                                                                                                                                                                    }                                            ⁢                                              (                                                                                                                                        V                              i                                                                                                            ≤                                                      2                            ⁢                                                                                                                            I                                  SS                                                                β                                                                                                                                    )                                            ⁢                                                                                           ⁢                                              (11a)                                                                                                                                                                                    I                      D1                                        =                                                                  1                        2                                            ⁢                                              I                        SS                                            ⁢                                              sgn                        ⁡                                                  (                                                      V                            i                                                    )                                                                    ⁢                                              (                                                                                                                                        V                              i                                                                                                            ≥                                                      2                            ⁢                                                                                                                            I                                  SS                                                                β                                                                                                                                    )                                            ⁢                                                                                           ⁢                                              (11b)                                                                                                                                                                 where,Vi=V1−V2  (12)Therefore, the circuit shown in FIG. 17 also functions as a voltage subtractor circuit. That is, the circuit shown in FIG. 17 is a voltage subtractor/adder circuit.
An explanation will now be made on a MOS differential amplifier circuit which has linear transconductance. FIG. 18 shows a general structure of this type of MOS differential amplifier circuit which is disclosed in Japanese patent laid-open publication No. 7-127887. The circuit of FIG. 18 comprises a MOS differential pair having MOS transistors M1 and M2 which are driven by a tail current Iss (=I0+βVi2/2).
Assuming that the MOS transistors are matched well, a differential output current ΔID=ID1−ID2 of the MOS differential pair comprising the transistors M1 and M2 becomes as follows:          {                                                                      Δ                ⁢                                                                   ⁢                                  I                  D                                            =                              β                ⁢                                                                   ⁢                                  V                  i                                ⁢                                                                                                    2                        ⁢                                                  I                          SS                                                                    β                                        -                                          V                      i                      2                                                                      ⁢                                  (                                                                                                          V                        i                                                                                    ≤                                                                                            I                          SS                                                β                                                                              )                                                      ⁢                                                                                       (                          13a                        )                                                                                          Δ                ⁢                                                                   ⁢                                  I                  D                                            =                                                I                  SS                                ⁢                                  sgn                  ⁡                                      (                                          V                      i                                        )                                                  ⁢                                  (                                                                                                          V                        i                                                                                    ≥                                                                                            I                          SS                                                β                                                                              )                                                      ⁢                                                                                       (                          13b                        )                              
Therefore, when the value within √{square root over ( )} in the formula (13a) is a constant value, the differential output current ΔID of the MOS differential pair becomes linear. That is, the condition of the tail current in an adaptive-biasing differential pair becomes as follows:                               I          SS                =                              I            0                    +                                    1              2                        ⁢            β            ⁢                                                   ⁢                          V              i              2                                                          (        14        )            
Therefore, by driving a MOS differential pair by using a tail current which has a square-law characteristic of an input voltage, it is possible to completely compensate transconductance of the MOS differential pair. The method of driving a MOS differential pair by using a current which varies dynamically such that the transconductance becomes linear is called an adaptive-biasing method. Also, the differential pair which has a linear transconductance obtained in this way is called an adaptive-biasing differential pair.
FIG. 19 shows an example of a concrete circuit of an adaptive-biasing differential pair in which a tail current is supplied thereto by using quadri-tail cell as a squaring circuit.
An output current IL of an output of the quadri-tail cell shown in FIG. 19 can be obtained as follows:                               I          L                =                              I            D3                    +                      I            D4                                              (        15        )                                {                                                                                                  I                    L                                    =                                                                                    I                        0                                            4                                        -                                                                                            β                          ⁢                                                                                                           ⁢                                                      V                            i                            2                                                                          4                                            ⁢                                              (                                                                                                                                        V                              i                                                                                                            ≤                                                                                                                    2                                ⁢                                                                  I                                  0                                                                                                                            3                                ⁢                                β                                                                                                                                    )                                                                                            ⁢                                                                                                                                                                                 I                    L                                    =                                                            2                      3                                        -                                                                  I                        0                                            ⁢                                                                                                    β                            ⁢                                                                                                                   ⁢                                                          V                              i                              2                                                                                +                                                      2                            ⁢                            β                            ⁢                                                                                                                   ⁢                                                                                                                        V                                i                                                                                                                    ⁢                                                                                          2                                ⁢                                                                  (                                                                                                                                                    6                                        ⁢                                                                                  I                                          0                                                                                                                    β                                                                        -                                                                          V                                      i                                      2                                                                                                        )                                                                                                                                                                    18                                            ⁢                                                                                           ⁢                                              (                                                  15b                                                )                                                                                            ⁢                                                                                                                                           (                                                                                                    2                        ⁢                                                  I                          0                                                                                            3                        ⁢                        β                                                                              ≤                                                                                V                      i                                                                            ≤                                      2                    ⁢                                                                                            I                          0                                                β                                                                                            )                                                                                                                                                    I                      L                                        =                                          0                      ⁢                                              (                                                                                                                                        V                              i                                                                                                            ≥                                                      2                            ⁢                                                                                                                            I                                  0                                                                β                                                                                                                                    )                                                                              )                                ⁢                                                                   ⁢                                  (                                      15c                                    )                                                                                        (                  15a                )            Therefore, it is possible to obtain a square-law current.
In order to adaptively bias a MOS differential pair by driving the MOS differential pair by using the output current of the quadri-tail cell, it is possible to set the tail current as determined by the following formula:Iss=2I0−2IL  (16)By setting the tail current in accordance with the above formula, transconductance becomes a constant value gm=√{(2I0)/β}, in a range of an input voltage |Vi|≦√{square root over ( )}{(2I0)/(3β)}.
A differential output current ΔI (=ID1−ID2) of an adaptive-biasing differential pair which uses the quadri-tail cell becomes as follows:                                  {                                                                                                                Δ                      ⁢                                                                                           ⁢                      I                                        =                                                                                                                        2                            ⁢                                                          I                              0                                                                                β                                                                    ⁢                                                                        V                          i                                                ⁡                                                  (                                                                                                                                                  V                                i                                                                                                                    ≤                                                                                                                            2                                  ⁢                                                                      I                                    0                                                                                                                                    3                                  ⁢                                  β                                                                                                                                              )                                                                                                      ⁢                                                                                                                                                                                   Δ                    ⁢                                                                                   ⁢                    I                                    =                                                                                    β                        ⁢                                                                                                   ⁢                                                  V                          i                                                                    3                                        ⁢                                                                                            12                          ⁢                                                                                    I                              0                                                        β                                                                          -                                                  7                          ⁢                                                      V                            i                            2                                                                          +                                                  4                          ⁢                                                                                                                V                              i                                                                                                            ⁢                                                                                    2                              ⁢                                                              (                                                                                                                                            6                                      ⁢                                                                              I                                        0                                                                                                              β                                                                    -                                                                      V                                    i                                    2                                                                                                  )                                                                                                                                                                          ⁢                                                                                   ⁢                                          (                                              17b                                            )                                                                                                                                            (                                                                                                              2                          ⁢                                                      I                            0                                                                                                    3                          ⁢                          β                                                                                      ≦                                                                                        V                        i                                                                                    ≦                                                                                            (                                                      1                            +                                                          1                                                              2                                                                                                              )                                                ⁢                                                                              I                            0                                                    β                                                                                                      )                                                                                                                          Δ                    ⁢                                                                                   ⁢                    I                                    =                                                                                                              β                          ⁢                                                                                                           ⁢                                                      V                            i                            2                                                                          +                                                  6                          ⁢                                                      I                            0                                                                          +                                                  2                          ⁢                          β                          ⁢                                                                                                                V                              i                                                                                                            ⁢                                                                                    2                              ⁢                                                              (                                                                                                                                            6                                      ⁢                                                                              I                                        0                                                                                                              β                                                                    -                                                                      V                                    i                                    2                                                                                                  )                                                                                                                                                        9                                        ⁢                                                                                   ⁢                                          (                                              17c                                            )                                                                                                                                            (                                                                                                              (                                                      1                            +                                                          1                                                              2                                                                                                              )                                                ⁢                                                                              I                            0                                                    β                                                                                      ≦                                                                                        V                        i                                                                                    ≦                                          2                      ⁢                                                                                                    I                            0                                                    β                                                                                                      )                                                                                                                          Δ                    ⁢                                                                                   ⁢                    I                                    =                                      2                    ⁢                                          I                      0                                        ⁢                                          sgn                      ⁡                                              (                                                  V                          i                                                )                                                              ⁢                                          (                                                                                                                              V                            i                                                                                                    ≥                                                                                                            I                              0                                                        β                                                                                              )                                        ⁢                                                                                   ⁢                                          (                                              17d                                            )                                                                                                                                (                      17a                    )                    Transconductance can be obtained by differentiating the formulas (17a) through (17d) by an input voltage Vi.                                  {                                                                                                                                        ⅆ                                                  (                                                      Δ                            ⁢                                                                                                                   ⁢                            I                                                    )                                                                                            ⅆ                                                  V                          i                                                                                      =                                                                                                                        2                            ⁢                                                          I                              0                                                                                β                                                                    ⁢                                              (                                                                                                                                        V                              i                                                                                                            ≤                                                                                                                    2                                ⁢                                                                  I                                  0                                                                                                                            3                                ⁢                                β                                                                                                                                    )                                                                              ⁢                                                                                                                                                                                                                                 ⅆ                                                  (                                                      Δ                            ⁢                                                                                                                   ⁢                            I                                                    )                                                                                            ⅆ                                                                                                   ⁢                                                  V                          i                                                                                      =                                                                                            β                          3                                                ⁢                                                                                                            12                              ⁢                                                                                                I                                  0                                                                β                                                                                      -                                                          7                              ⁢                                                              V                                i                                2                                                                                      +                                                          4                              ⁢                                                                                                                                V                                  i                                                                                                                            ⁢                                                                                                2                                  ⁢                                                                      (                                                                                                                                                            6                                          ⁢                                                                                      I                                            0                                                                                                                          β                                                                            -                                                                              V                                        i                                        2                                                                                                              )                                                                                                                                                                                                                          +                                                        ⁢                                                                                                                                                                                                         β                      3                                        ⁢                                          V                      i                                        ⁢                                          {                                                                                                    -                            7                                                    ⁢                                                      V                            i                                                                          +                                                  2                          ⁢                                                      sgn                            ⁡                                                          (                                                              V                                i                                                            )                                                                                ⁢                                                                                    2                              ⁢                                                              (                                                                                                                                            6                                      ⁢                                                                              I                                        0                                                                                                              β                                                                    -                                                                      V                                    i                                    2                                                                                                  )                                                                                                                                    -                                                                              4                            ⁢                            β                            ⁢                                                                                                                        V                                i                                                                                                                    ⁢                                                          V                              i                                                                                                                                          2                              ⁢                                                              (                                                                                                                                            6                                      ⁢                                                                              I                                        0                                                                                                              β                                                                    -                                                                      V                                    i                                    2                                                                                                  )                                                                                                                                                        }                                                        ⁢                                                                                                                                                                                   1                                                                                                                        12                            ⁢                                                          I                              0                                                                                β                                                -                                                  7                          ⁢                                                      V                            i                            2                                                                          +                                                  4                          ⁢                                                                                                                V                              i                                                                                                            ⁢                                                                                    2                              ⁢                                                              (                                                                                                                                            6                                      ⁢                                                                              I                                        0                                                                                                              β                                                                    -                                                                      V                                    i                                    2                                                                                                  )                                                                                                                                                                                          ⁢                                                                           ⁢                                      (                                          18b                                        )                                                                                                                        (                                                                                                              2                          ⁢                                                      I                            0                                                                                                    3                          ⁢                          β                                                                    ≤                                                                                                V                          i                                                                                            ≤                                                                                                    (                                                          1                              +                                                              1                                                                  2                                                                                                                      )                                                    ⁢                                                                                    I                              0                                                        β                                                                                                                                )                                                                                                                                                ⅆ                                              (                                                  Δ                          ⁢                                                                                                           ⁢                          I                                                )                                                                                    ⅆ                                              V                        i                                                                              =                                                                                    2                        ⁢                        β                                            9                                        ⁢                                          {                                                                        V                          i                                                +                                                                              sgn                            ⁡                                                          (                                                              V                                i                                                            )                                                                                ⁢                                                                                    2                              ⁢                                                              (                                                                                                                                            6                                      ⁢                                                                              I                                        0                                                                                                              β                                                                    -                                                                      V                                    i                                    2                                                                                                  )                                                                                                                                    -                                                                                                   ⁢                                                  (                                                      18c                                                    )                                                                                                                                                                                                                                                            4                        ⁢                        β                        ⁢                                                                                                        V                            i                                                                                                    ⁢                                                  V                          i                                                                                                                      2                          ⁢                                                      (                                                                                                                            6                                  ⁢                                                                      I                                    0                                                                                                  β                                                            -                                                              V                                i                                2                                                                                      )                                                                                                                }                                    ⁢                                      (                                                                                                                        (                                                          1                              +                                                              1                                                                  2                                                                                                                      )                                                    ⁢                                                                                                           ⁢                                                                                    I                              0                                                        β                                                                                              ≤                                                                                                V                          i                                                                                            ≤                                                                                                    I                            0                                                    β                                                                                      )                                                                                                                                                                                          ⅆ                                                  (                                                      Δ                            ⁢                                                                                                                   ⁢                            I                                                    )                                                                    ⁢                                                                                                                                   ⅆ                                              V                        i                                                                              =                                      0                    ⁢                                          (                                                                                                                              V                            i                                                                                                    ≥                                                  2                          ⁢                                                                                                                    I                                0                                                            β                                                                                                                          )                                        ⁢                                                                                   ⁢                                          (                                              18d                                            )                                                                                                                                (                      18a                    )                    
As can be seen from the above formulas, the transconductance of an adaptive-biasing differential pair becomes a constant value gm=√{square root over ( )}{(2I0)/β}, that is, the transconductance of an adaptive-biasing differential pair shows a flat characteristic, in a range of an input voltage |Vi|≦√{square root over ( )}{(2I0)/(3β)}.
Although the above-mentioned voltage adder circuit has both the subtraction function and addition function, the subtraction function is inferior in linearity to the addition function.
Also, in a linear transconductance amplifier for realizing both the subtraction function and addition function, an input voltage range in which the linear transconductance amplifier operates linearly depends on an input voltage range within which the squaring circuit for supplying the tail current has a square-law characteristic. However, it is difficult to realize a squaring circuit which has an input voltage range within which the squaring circuit shows a square-law characteristic throughout the whole operating input voltage range of the MOS differential pair. Therefore, conventionally, it was impossible to realize a linear transconductance amplifier having a wide linear input voltage range.
In the field of analog signal processing, a circuit for performing subtraction and/or addition is an essential function block. Especially, the requirement for realizing a MOS differential amplifier circuit having linear subtraction and addition function has become stronger.
Also, such MOS differential amplifier circuit having linear subtraction and addition function can be realized by using a differential amplifier circuit having linear transconductance. Therefore, such differential amplifier circuit having linear transconductance is also an essential function block in the field of analog signal processing. Especially, the requirement for realizing a MOS differential amplifier circuit having linear transconductance has become stronger.